Finally, this got posted! All inhabitants claim to tell the truth. Suppose the inhabitant said, "Either I'm a day-knight or it is day, but not both." That statement means "Either I'm a day-knight and it is night, or I'm a night-knight and it is day." That means "I am lying," which is a paradox. Therefore, she said, "Either I'm a night-knight and it is day, but not both." She lied when she said, "I said I'm a day-knight!" Therefore, the last answer was also a lie, so she is a night-knight. Since night-knights lie in the day, it is day.